Problem 26 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f/g)(0)\)

Step-by-Step Solution

Answer

Therefore, the value of \((f/g)(0)\) is -0.25.

Step 1: Determine the Value of f(0)

First, find the value of the function f at point 0 by substituting x with 0 in the equation for f. That gives \(f(0) = (0)^2 + 1 = 1\).

Step 2: Determine the Value of g(0)

Next, find the value of the function g at point 0 by substituting x with 0 in the equation for g. This results in \(g(0) = 0 - 4 = -4\).

Step 3: Evaluate (f/g)(0)

Finally, evaluate the function \((f/g)(0)\) by dividing the value of f at 0 by the value of g at 0, that is \(1 / -4 = -0.25\).

Key Concepts

Evaluate Functions

To understand the concept of evaluating functions, imagine you have a machine that transforms any number you input into another number according to a set rule. This rule is represented by the function. When you evaluate a function like f(x) for a specific value, you're essentially plugging that value into the rule to see what number comes out.

For instance, if f(x) = x² + 1, and you want to evaluate f(0), you replace every x in the equation with 0.

• Start: f(x) = x² + 1
• Substitute: f(0) = (0)² + 1
• Simplify: f(0) = 0 + 1
• Result: f(0) = 1

This process shows us that evaluating the function f at 0 yields the number 1. Evaluation is just following the function's rule with a specific input.

Function Operations

Function operations are ways to combine two or more functions to create a new function. There are four basic operations: addition, subtraction, multiplication, and division. In our exercise, the operation is division, so we're interested in the function (f/g)(x).

To find (f/g)(x), you simply divide the output of f at any x by the output of g at the same x. But remember, you cannot divide by zero as it's undefined. So the function (f/g)(x) is only defined for values of x where g(x) ≠ 0.

In our case, since g(0) = -4, it's safe to divide by g(0) because it's not zero. Thus, (f/g)(0) is valid and can be calculated.

Substituting Values

Substituting values into functions is a critical skill that allows us to evaluate functions at particular points. This step involves replacing the variable in the function, usually x, with a specific value.

For example, if you need to evaluate g(x) = x - 4 for x=0:

• Start: g(x) = x - 4
• Substitute: g(0) = 0 - 4
• Simplify: g(0) = -4

This substitution gives us the specific output for g when x is 0. Whenever you substitute a value into a function, make sure to perform any necessary arithmetic operations that follow from the substitution to find the correct result. Substitution is used not only for evaluating functions but also while performing function operations.